Logarithms - Useful Background to Understand Power Laws in Biological Scaling

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Biological Scaling:  Importance of Size:  Laws of Physics  determine how animals are formed and how they function. Animal bodies are designed to operate under the constraints set by the laws of physics.  Strength of animal bones will determine the maximum mass it can have.  How efficiently heat produced by metabolic activity inside the body may be eliminated, how oxygen is transferred to the blood - these processes have limits set by physics and determine the shape and size and methods that animals use to optimize their bodies.

What is amazing is that many body functions of animals, like metabolic activity, heart size, oxygen consumption etc. follow general laws expressed as some power of their mass, and the same power laws hold over a very wide size range from the tiny shrew to the massive whale.  

I intend to look at biological scaling laws.

It can be sometimes difficult to interpret power laws as they are non-linear functions. Power laws may be expressed in an 'easier to interpret' linear form by using logarithms. Logarithms are not somethings that everybody is totally comfortable with and a short primer on logs might be useful before discussing the subject of scaling laws in more detail.  

In mathematics, we define a function as an operation that changes the value, in a well-defined way, of the quantity it operates on.  For example:

square root is a function is                    sqrt(9) = 3 

tan is a trigonometric function               tan(60) = 1.732

Most frequently used functions are found as hot-buttons on your scientific calculator.

Similarly logarithm is a function, also found as a hot-button (log) on your calculator. 

Logs are useful as they change products to sums, and divisions to differences

                                   log (AB)  = log A + log B

                                   log (A/B) = log A – log B

For a power relation: 
       log (A³) = log (A.A.A) = log A + log A + log A = 3 log A

If                                y = xⁿ            then        log y = n log x

n does not have to be an integer.  It can also be a positive or negative number.

In the case of a power law of the form y = a xⁿ       

we have   log y = log (a xⁿ)  = log a + log (xⁿ) = log a + n log x

You might notice that                        log y = log a + n log x               
is the equation of a straight line when one plots log y along the y-axis and log x along the x-axis.  Thus we have converted a power relation which plots a curve, to a linear relation that will plot a straight line.  Also notice that the slope of the straight line is the exponent n of the power relation and the intercept on the y-axis is log(a). 

The log functions we shall use are to the base 10.  This is what the scientific calculator gives too.   This means that log10 = 1 (you can check it on your calculator)

To understand it better: 

Suppose you take log of number 2. 

It will be a number p such that 2 = 10^p 

Taking logs gives us           log 2 = log (10^p)  = p log10 = p                            since log10 = 1

Using your calculator you will find that p = 0.30103

So you see, logarithms are just another way of writing numbers.  In special situations, the type we shall deal with, logs make the equations simpler to interpret.

In the following, I show how logs change power relations to linear relations:   The XCEL worksheet shows what is plotted in the two graphs that follow: 

(please click on the slide to see its bigger image)

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